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Author: William A. Stein
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Dear William,
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I have begun to read your thesis, and-- as you can expect-- I am very
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impressed with it! Probably, for table 1.1 instead of the cryptic
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parenthesis "(ignoring odd parts)" it might be good to expand the
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parenthesis a bit by saying something like
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(the entries in the columns "order of Sha" and "modular degree" are only
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the odd parts of the actual "order of Sha" and "modular degree").
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In table 1.1 do you know the Mordell-Weil ranks of the abelian varieties in
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the B column? I would imagine that the Mordell-Weil groups of these B's,
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say when tensored with the field of fractions F of the corresponding ring
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of Fourier coefficients, is always of dimension 2 over F. If not, e.g., if
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some of them are of higher dimension, it would be worth remarking.
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Also, since you talk about reversing "visiblity", i.e.,since you sometimes
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start with an abelian variety (of "type B", to follow the notation of your
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table, i.e.) an abelian variety B with positive Mordell-Weil rank to try
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to guarantee some nontrivial Sha in an abelian variety A which is congruent
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to B, one might "more systematically" try to follow through on this: Do you
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have data on how often it is the case, for example, that when you have an
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optimal abelian variety B with MW-rank 2 (over F, of course), you also get
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an A congruent to B, with some Sha in A made visible in B? Must be very
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very often, I imagine (especially when the level goes up). I haven't
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found a place where you explicitly list this information.
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The example of level 2333 is amazing! Clearly this example is sitting
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around waiting for someone to "do something" with it... but what?
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The cases 1483, 1567, 2029, and 2593? are also interesting because, as
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I gather from the table (perhaps you should say explicitly whether this is
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what happens...) ALL of Sha (even though it has two nontrivial p-primary
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components in each of these cases) is made visible in a single B. I assume
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that this is true? In contrast in the case 1913E I guess it can only be
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the 5-primary component of Sha that is visible in 1913A (?) In any event,
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all this should be discussed more explicitly (unless yuou do this somewhere
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and I ahve missed it).
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More later,
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Barry
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